Problem: Solve for $n$, if $9^n\cdot9^n\cdot9^n\cdot9^n=81^4$.
Solution: The equation, $9^n\cdot9^n\cdot9^n\cdot9^n=81^4$, can be written as $9^{4n}=81^4$. We also know that $81=9^2$, so we can rewrite the equation as $9^{4n}=9^{2(4)}$.  Solving for $n$, gives $n=\boxed{2}$.